Construction of finite-dimensional nonlinear filter via ODEs

Consider the robust Duncan-Mortensen-Zakai (DMZ) equation arising from Yau filtering system which includes Kalman-Bucy filtering system and Benes filtering. The main problem of nonlinear filtering is to solve this robust DMZ equation in real time. It is shown that this equation can be solved explicitly with an arbitrary initial condition by solving a linear system of ODEs and a Kolmogorov-type equation. Furthermore, it is shown that the Kolmogorov-type equation can be solved via the Riccati-type system of ODEs. Thus the robust DMZ equation arising from Yau's filtering system is shown to be solvable in real time.